Continuous linear maps #
In this file we define the type of continuous (semi)linear maps between topological modules that are continuous, and endow it with its algebraic structure.
Later files endow it with a topological structure, see the docstring of
Mathlib/Topology/Algebra/Module/Spaces/ContinuousLinearMap.lean.
Main definitions #
ContinuousLinearMapis the type of (semi)linear maps between two topological modules that are continuous. It is denoted byM →L[R] Nin theR-linear case,M →SL[σ] Nin theσ-semilinear case, andM →L⋆[R] Nin the conjugate-linear (antilinear) case.StrongDual R Mis an abbreviation forM →L[R] R, the type of continuousR-linear forms onM. As a vector space, it is often called the "topological dual ofM". We use the name "strong dual" because it will (in later files) be endowed with the strong-dual topology, namely the topology of uniform convergence on bounded subsets.ContinuousLinearMap.addCommMonoid,ContinuousLinearMap.module,... : the algebraic structures onM →SL[σ] N.
Notation #
M →L[R] N: the type ofR-linear continuous maps fromMtoN;M →SL[σ] N: the type ofσ-semilinear continuous maps fromMtoN;M →L⋆[σ] N: the type of conjugate-linear (antilinear) continuous maps fromMtoN;f ∘L g: the composition of two continuous linear maps;f ∘SL g: the composition of two continuous semilinear maps.
structure
ContinuousLinearMap
{R : Type u_1}
{S : Type u_2}
[Semiring R]
[Semiring S]
(σ : R →+* S)
(M : Type u_3)
[TopologicalSpace M]
[AddCommMonoid M]
(M₂ : Type u_4)
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R M]
[Module S M₂]
extends M →ₛₗ[σ] M₂ :
Type (max u_3 u_4)
Continuous linear maps between modules. We only put the type classes that are necessary for the
definition, although in applications M and M₂ will be topological modules over the topological
ring R.
- toFun : M → M₂
- cont : Continuous (↑self).toFun
Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications
MandM₂will be topological modules over the topological ringR.
Instances For
Continuous linear maps between modules. We only put the type classes that are necessary for the
definition, although in applications M and M₂ will be topological modules over the topological
ring R.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Continuous linear maps between modules. We only put the type classes that are necessary for the
definition, although in applications M and M₂ will be topological modules over the topological
ring R.
Equations
- One or more equations did not get rendered due to their size.
Instances For
class
ContinuousSemilinearMapClass
(F : Type u_1)
{R : outParam (Type u_2)}
{S : outParam (Type u_3)}
[Semiring R]
[Semiring S]
(σ : outParam (R →+* S))
(M : outParam (Type u_4))
[TopologicalSpace M]
[AddCommMonoid M]
(M₂ : outParam (Type u_5))
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R M]
[Module S M₂]
[FunLike F M M₂]
extends SemilinearMapClass F σ M M₂, ContinuousMapClass F M M₂ :
ContinuousSemilinearMapClass F σ M M₂ asserts F is a type of bundled continuous
σ-semilinear maps M → M₂. See also ContinuousLinearMapClass F R M M₂ for the case where
σ is the identity map on R. A map f between an R-module and an S-module over a ring
homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y
and f (c • x) = (σ c) • f x.
Instances
@[reducible, inline]
abbrev
ContinuousLinearMapClass
(F : Type u_1)
(R : outParam (Type u_2))
[Semiring R]
(M : outParam (Type u_3))
[TopologicalSpace M]
[AddCommMonoid M]
(M₂ : outParam (Type u_4))
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R M]
[Module R M₂]
[FunLike F M M₂]
:
ContinuousLinearMapClass F R M M₂ asserts F is a type of bundled continuous
R-linear maps M → M₂. This is an abbreviation for
ContinuousSemilinearMapClass F (RingHom.id R) M M₂.
Equations
- ContinuousLinearMapClass F R M M₂ = ContinuousSemilinearMapClass F (RingHom.id R) M M₂
Instances For
@[reducible, inline]
abbrev
StrongDual
(R : Type u_1)
[Semiring R]
[TopologicalSpace R]
(M : Type u_2)
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
:
Type (max u_1 u_2)
The strong dual of a topological vector space M over a ring R. This is the space of
continuous linear functionals and is equipped with the topology of uniform convergence
on bounded subsets. StrongDual R M is an abbreviation for M →L[R] R.
Equations
- StrongDual R M = (M →L[R] R)
Instances For
Properties that hold for non-necessarily commutative semirings. #
@[implicit_reducible]
instance
ContinuousLinearMap.LinearMap.coe
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
Coerce continuous linear maps to linear maps.
Equations
theorem
ContinuousLinearMap.coe_injective
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
@[implicit_reducible]
instance
ContinuousLinearMap.funLike
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
instance
ContinuousLinearMap.continuousSemilinearMapClass
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
ContinuousSemilinearMapClass (M₁ →SL[σ₁₂] M₂) σ₁₂ M₁ M₂
theorem
ContinuousLinearMap.coe_mk
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →ₛₗ[σ₁₂] M₂)
(h : Continuous f.toFun)
:
@[simp]
theorem
ContinuousLinearMap.coe_mk'
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →ₛₗ[σ₁₂] M₂)
(h : Continuous f.toFun)
:
theorem
ContinuousLinearMap.continuous
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
Continuous ⇑f
theorem
ContinuousLinearMap.continuous_toLinearMap
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
Continuous ⇑↑f
@[simp]
theorem
ContinuousLinearMap.uniformContinuous
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{E₁ : Type u_9}
{E₂ : Type u_10}
[UniformSpace E₁]
[UniformSpace E₂]
[AddCommGroup E₁]
[AddCommGroup E₂]
[Module R₁ E₁]
[Module R₂ E₂]
[IsUniformAddGroup E₁]
[IsUniformAddGroup E₂]
(f : E₁ →SL[σ₁₂] E₂)
:
@[simp]
theorem
ContinuousLinearMap.coe_inj
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{f g : M₁ →SL[σ₁₂] M₂}
:
theorem
ContinuousLinearMap.coeFn_injective
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_injective
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
@[simp]
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_inj
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{f g : M₁ →SL[σ₁₂] M₂}
:
def
ContinuousLinearMap.Simps.apply
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(h : M₁ →SL[σ₁₂] M₂)
:
M₁ → M₂
See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.
Equations
Instances For
def
ContinuousLinearMap.Simps.coe
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(h : M₁ →SL[σ₁₂] M₂)
:
See Note [custom simps projection].
Equations
Instances For
theorem
ContinuousLinearMap.ext
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{f g : M₁ →SL[σ₁₂] M₂}
(h : ∀ (x : M₁), f x = g x)
:
theorem
ContinuousLinearMap.ext_iff
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{f g : M₁ →SL[σ₁₂] M₂}
:
@[simp]
theorem
ContinuousLinearMap.coe_coe
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
def
ContinuousLinearMap.copy
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
(f' : M₁ → M₂)
(h : f' = ⇑f)
:
Copy of a ContinuousLinearMap with a new toFun equal to the old one. Useful to fix
definitional equalities.
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_copy
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
(f' : M₁ → M₂)
(h : f' = ⇑f)
:
theorem
ContinuousLinearMap.copy_eq
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
(f' : M₁ → M₂)
(h : f' = ⇑f)
:
theorem
ContinuousLinearMap.range_coeFn_eq
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
theorem
ContinuousLinearMap.range_toLinearMap
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.map_zero
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.map_add
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
(x y : M₁)
:
@[simp]
theorem
ContinuousLinearMap.map_smulₛₗ
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
(c : R₁)
(x : M₁)
:
theorem
ContinuousLinearMap.map_smul
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₁ M₂]
(f : M₁ →L[R₁] M₂)
(c : R₁)
(x : M₁)
:
@[simp]
theorem
ContinuousLinearMap.map_smul_of_tower
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{R : Type u_9}
{S : Type u_10}
[Semiring S]
[SMul R M₁]
[Module S M₁]
[SMul R M₂]
[Module S M₂]
[LinearMap.CompatibleSMul M₁ M₂ R S]
(f : M₁ →L[S] M₂)
(c : R)
(x : M₁)
:
theorem
ContinuousLinearMap.ext_ring
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
{f g : R₁ →L[R₁] M₁}
(h : f 1 = g 1)
:
theorem
ContinuousLinearMap.ext_ring_iff
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
{f g : R₁ →L[R₁] M₁}
:
@[simp]
theorem
ContinuousLinearMap.apply_val_ker
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
(x : ↥(↑f).ker)
:
theorem
ContinuousLinearMap.eqOn_closure_span
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[T2Space M₂]
{s : Set M₁}
{f g : M₁ →SL[σ₁₂] M₂}
(h : Set.EqOn (⇑f) (⇑g) s)
:
Set.EqOn (⇑f) (⇑g) (closure ↑(Submodule.span R₁ s))
If two continuous linear maps are equal on a set s, then they are equal on the closure
of the Submodule.span of this set.
theorem
ContinuousLinearMap.ext_on
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[T2Space M₂]
{s : Set M₁}
(hs : Dense ↑(Submodule.span R₁ s))
{f g : M₁ →SL[σ₁₂] M₂}
(h : Set.EqOn (⇑f) (⇑g) s)
:
If the submodule generated by a set s is dense in the ambient module, then two continuous
linear maps equal on s are equal.
theorem
Submodule.topologicalClosure_map
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[RingHomSurjective σ₁₂]
[TopologicalSpace R₁]
[TopologicalSpace R₂]
[ContinuousSMul R₁ M₁]
[ContinuousAdd M₁]
[ContinuousSMul R₂ M₂]
[ContinuousAdd M₂]
(f : M₁ →SL[σ₁₂] M₂)
(s : Submodule R₁ M₁)
:
Under a continuous linear map, the image of the TopologicalClosure of a submodule is
contained in the TopologicalClosure of its image.
theorem
Submodule.topologicalClosure_mem_invtSubmodule
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
[ContinuousAdd M₁]
{f : M₁ →L[R₁] M₁}
{s : Submodule R₁ M₁}
(hs : s ∈ Module.End.invtSubmodule ↑f)
:
If a continuous linear map stabilizes a submodule, then it stabilizes its topological closure.
theorem
DenseRange.topologicalClosure_map_submodule
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[RingHomSurjective σ₁₂]
[TopologicalSpace R₁]
[TopologicalSpace R₂]
[ContinuousSMul R₁ M₁]
[ContinuousAdd M₁]
[ContinuousSMul R₂ M₂]
[ContinuousAdd M₂]
{f : M₁ →SL[σ₁₂] M₂}
(hf' : DenseRange ⇑f)
{s : Submodule R₁ M₁}
(hs : s.topologicalClosure = ⊤)
:
Under a dense continuous linear map, a submodule whose TopologicalClosure is ⊤ is sent to
another such submodule. That is, the image of a dense set under a map with dense range is dense.
@[implicit_reducible]
instance
ContinuousLinearMap.instSMul
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{S₂ : Type u_9}
[DistribSMul S₂ M₂]
[SMulCommClass R₂ S₂ M₂]
[ContinuousConstSMul S₂ M₂]
:
theorem
ContinuousLinearMap.smul_apply
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{S₂ : Type u_9}
[DistribSMul S₂ M₂]
[SMulCommClass R₂ S₂ M₂]
[ContinuousConstSMul S₂ M₂]
(c : S₂)
(f : M₁ →SL[σ₁₂] M₂)
(x : M₁)
:
@[simp]
theorem
ContinuousLinearMap.coe_smul
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{S₂ : Type u_9}
[DistribSMul S₂ M₂]
[SMulCommClass R₂ S₂ M₂]
[ContinuousConstSMul S₂ M₂]
(c : S₂)
(f : M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.coe_smul'
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{S₂ : Type u_9}
[DistribSMul S₂ M₂]
[SMulCommClass R₂ S₂ M₂]
[ContinuousConstSMul S₂ M₂]
(c : S₂)
(f : M₁ →SL[σ₁₂] M₂)
:
instance
ContinuousLinearMap.isScalarTower
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{S₂ : Type u_9}
{T₂ : Type u_10}
[DistribSMul S₂ M₂]
[SMulCommClass R₂ S₂ M₂]
[ContinuousConstSMul S₂ M₂]
[DistribSMul T₂ M₂]
[SMulCommClass R₂ T₂ M₂]
[ContinuousConstSMul T₂ M₂]
[SMul S₂ T₂]
[IsScalarTower S₂ T₂ M₂]
:
IsScalarTower S₂ T₂ (M₁ →SL[σ₁₂] M₂)
instance
ContinuousLinearMap.smulCommClass
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{S₂ : Type u_9}
{T₂ : Type u_10}
[DistribSMul S₂ M₂]
[SMulCommClass R₂ S₂ M₂]
[ContinuousConstSMul S₂ M₂]
[DistribSMul T₂ M₂]
[SMulCommClass R₂ T₂ M₂]
[ContinuousConstSMul T₂ M₂]
[SMulCommClass S₂ T₂ M₂]
:
SMulCommClass S₂ T₂ (M₁ →SL[σ₁₂] M₂)
@[implicit_reducible]
instance
ContinuousLinearMap.mulAction
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{S₂ : Type u_9}
[Monoid S₂]
[DistribMulAction S₂ M₂]
[SMulCommClass R₂ S₂ M₂]
[ContinuousConstSMul S₂ M₂]
:
Equations
- ContinuousLinearMap.mulAction = { toSMul := ContinuousLinearMap.instSMul, mul_smul := ⋯, one_smul := ⋯ }
@[implicit_reducible]
instance
ContinuousLinearMap.zero
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
The continuous map that is constantly zero.
@[implicit_reducible]
instance
ContinuousLinearMap.inhabited
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
Equations
- ContinuousLinearMap.inhabited = { default := 0 }
@[simp]
theorem
ContinuousLinearMap.default_def
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
@[simp]
theorem
ContinuousLinearMap.zero_apply
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(x : M₁)
:
@[simp]
theorem
ContinuousLinearMap.coe_zero
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
theorem
ContinuousLinearMap.coe_zero'
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
@[simp]
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_zero
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
:
@[implicit_reducible]
instance
ContinuousLinearMap.uniqueOfLeft
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[Subsingleton M₁]
:
Equations
@[implicit_reducible]
instance
ContinuousLinearMap.uniqueOfRight
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[Subsingleton M₂]
:
Equations
theorem
ContinuousLinearMap.exists_ne_zero
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
{f : M₁ →SL[σ₁₂] M₂}
(hf : f ≠ 0)
:
∃ (x : M₁), f x ≠ 0
def
ContinuousLinearMap.id
(R₁ : Type u_1)
[Semiring R₁]
(M₁ : Type u_4)
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
the identity map as a continuous linear map.
Equations
- ContinuousLinearMap.id R₁ M₁ = { toLinearMap := LinearMap.id, cont := ⋯ }
Instances For
@[implicit_reducible]
instance
ContinuousLinearMap.one
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
Equations
- ContinuousLinearMap.one = { one := ContinuousLinearMap.id R₁ M₁ }
theorem
ContinuousLinearMap.one_def
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
theorem
ContinuousLinearMap.id_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(x : M₁)
:
@[simp]
theorem
ContinuousLinearMap.coe_id
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
@[simp]
theorem
ContinuousLinearMap.coe_id'
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
@[simp]
theorem
ContinuousLinearMap.coe_one
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
@[simp]
theorem
ContinuousLinearMap.mk_id
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
@[simp]
theorem
ContinuousLinearMap.mk_one
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
@[simp]
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_id
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
@[simp]
theorem
ContinuousLinearMap.coe_eq_id
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{f : M₁ →L[R₁] M₁}
:
@[simp]
theorem
ContinuousLinearMap.one_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(x : M₁)
:
instance
ContinuousLinearMap.instNontrivialId
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[Nontrivial M₁]
:
Nontrivial (M₁ →L[R₁] M₁)
@[implicit_reducible]
instance
ContinuousLinearMap.add
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
:
@[simp]
theorem
ContinuousLinearMap.add_apply
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
(f g : M₁ →SL[σ₁₂] M₂)
(x : M₁)
:
@[simp]
theorem
ContinuousLinearMap.coe_add
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
(f g : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.coe_add'
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
(f g : M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_add
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
(f g : M₁ →SL[σ₁₂] M₂)
:
@[implicit_reducible]
instance
ContinuousLinearMap.instAddMonoid
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
:
Equations
- One or more equations did not get rendered due to their size.
@[implicit_reducible]
instance
ContinuousLinearMap.addCommMonoid
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
:
AddCommMonoid (M₁ →SL[σ₁₂] M₂)
Equations
- ContinuousLinearMap.addCommMonoid = { toAddMonoid := ContinuousLinearMap.instAddMonoid, add_comm := ⋯ }
@[simp]
theorem
ContinuousLinearMap.coe_sum
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
{ι : Type u_9}
(t : Finset ι)
(f : ι → M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.coe_sum'
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
{ι : Type u_9}
(t : Finset ι)
(f : ι → M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.sum_apply
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[ContinuousAdd M₂]
{ι : Type u_9}
(t : Finset ι)
(f : ι → M₁ →SL[σ₁₂] M₂)
(b : M₁)
:
def
ContinuousLinearMap.comp
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(g : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
Composition of continuous linear maps.
Instances For
Composition of continuous linear maps.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Composition of continuous linear maps.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_comp
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(h : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.coe_comp'
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(h : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_comp
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(h : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.comp_apply
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(g : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
(x : M₁)
:
@[simp]
theorem
ContinuousLinearMap.comp_id
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.id_comp
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.leftInverse_of_comp
{R : Type u_9}
{E : Type u_10}
{F : Type u_11}
[Semiring R]
[TopologicalSpace E]
[AddCommMonoid E]
[Module R E]
[TopologicalSpace F]
[AddCommMonoid F]
[Module R F]
{f : E →L[R] F}
{g : F →L[R] E}
(hinv : g ∘SL f = ContinuousLinearMap.id R E)
:
Function.LeftInverse ⇑g ⇑f
g ∘ f = id as ContinuousLinearMaps implies g ∘ f = id as functions.
theorem
ContinuousLinearMap.rightInverse_of_comp
{R : Type u_9}
{E : Type u_10}
{F : Type u_11}
[Semiring R]
[TopologicalSpace E]
[AddCommMonoid E]
[Module R E]
[TopologicalSpace F]
[AddCommMonoid F]
[Module R F]
{f : E →L[R] F}
{g : F →L[R] E}
(hinv : f ∘SL g = ContinuousLinearMap.id R F)
:
Function.RightInverse ⇑g ⇑f
f ∘ g = id as ContinuousLinearMaps implies f ∘ g = id as functions.
@[simp]
theorem
ContinuousLinearMap.comp_zero
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(g : M₂ →SL[σ₂₃] M₃)
:
@[simp]
theorem
ContinuousLinearMap.zero_comp
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(f : M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.comp_add
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[ContinuousAdd M₂]
[ContinuousAdd M₃]
(g : M₂ →SL[σ₂₃] M₃)
(f₁ f₂ : M₁ →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.add_comp
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[ContinuousAdd M₃]
(g₁ g₂ : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.comp_finsetSum
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
{ι : Type u_9}
{s : Finset ι}
[ContinuousAdd M₂]
[ContinuousAdd M₃]
(g : M₂ →SL[σ₂₃] M₃)
(f : ι → M₁ →SL[σ₁₂] M₂)
:
@[deprecated ContinuousLinearMap.comp_finsetSum (since := "2026-04-08")]
theorem
ContinuousLinearMap.comp_finset_sum
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
{ι : Type u_9}
{s : Finset ι}
[ContinuousAdd M₂]
[ContinuousAdd M₃]
(g : M₂ →SL[σ₂₃] M₃)
(f : ι → M₁ →SL[σ₁₂] M₂)
:
Alias of ContinuousLinearMap.comp_finsetSum.
theorem
ContinuousLinearMap.finsetSum_comp
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
{ι : Type u_9}
{s : Finset ι}
[ContinuousAdd M₃]
(g : ι → M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
@[deprecated ContinuousLinearMap.finsetSum_comp (since := "2026-04-08")]
theorem
ContinuousLinearMap.finset_sum_comp
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
{ι : Type u_9}
{s : Finset ι}
[ContinuousAdd M₃]
(g : ι → M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
Alias of ContinuousLinearMap.finsetSum_comp.
theorem
ContinuousLinearMap.comp_assoc
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
{M₄ : Type u_8}
[TopologicalSpace M₄]
[AddCommMonoid M₄]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
{R₄ : Type u_9}
[Semiring R₄]
[Module R₄ M₄]
{σ₁₄ : R₁ →+* R₄}
{σ₂₄ : R₂ →+* R₄}
{σ₃₄ : R₃ →+* R₄}
[RingHomCompTriple σ₁₃ σ₃₄ σ₁₄]
[RingHomCompTriple σ₂₃ σ₃₄ σ₂₄]
[RingHomCompTriple σ₁₂ σ₂₄ σ₁₄]
(h : M₃ →SL[σ₃₄] M₄)
(g : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.cancel_left
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
{g : M₂ →SL[σ₂₃] M₃}
{f₁ f₂ : M₁ →SL[σ₁₂] M₂}
(hg : Function.Injective ⇑g)
(h : g ∘SL f₁ = g ∘SL f₂)
:
theorem
ContinuousLinearMap.cancel_left'
{R₁ : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R₁]
[Semiring R₂]
[Semiring R₃]
{σ₁₂ : R₁ →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R₁ →+* R₃}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{M₃ : Type u_7}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₁ M₁]
[Module R₂ M₂]
[Module R₃ M₃]
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
{g : M₂ →SL[σ₂₃] M₃}
{f₁ f₂ : M₁ →SL[σ₁₂] M₂}
(hg : Function.Injective ⇑g)
:
@[implicit_reducible]
instance
ContinuousLinearMap.instMul
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
Equations
theorem
ContinuousLinearMap.mul_def
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(f g : M₁ →L[R₁] M₁)
:
@[simp]
theorem
ContinuousLinearMap.coe_mul
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(f g : M₁ →L[R₁] M₁)
:
@[simp]
theorem
ContinuousLinearMap.coe_mul'
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(f g : M₁ →L[R₁] M₁)
:
theorem
ContinuousLinearMap.mul_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(f g : M₁ →L[R₁] M₁)
(x : M₁)
:
@[implicit_reducible]
instance
ContinuousLinearMap.monoidWithZero
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
:
MonoidWithZero (M₁ →L[R₁] M₁)
Equations
- One or more equations did not get rendered due to their size.
@[simp]
theorem
ContinuousLinearMap.coe_pow'
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(f : M₁ →L[R₁] M₁)
(n : ℕ)
:
@[simp]
theorem
ContinuousLinearMap.coe_pow
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(f : M₁ →L[R₁] M₁)
(n : ℕ)
:
@[implicit_reducible]
instance
ContinuousLinearMap.instNatCast
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
@[implicit_reducible]
instance
ContinuousLinearMap.semiring
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
Equations
- One or more equations did not get rendered due to their size.
def
ContinuousLinearMap.toLinearMapRingHom
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
ContinuousLinearMap.toLinearMap as a RingHom.
Equations
- ContinuousLinearMap.toLinearMapRingHom = { toFun := ContinuousLinearMap.toLinearMap, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
Instances For
@[simp]
theorem
ContinuousLinearMap.toLinearMapRingHom_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
(self : M₁ →L[R₁] M₁)
:
@[simp]
theorem
ContinuousLinearMap.natCast_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
(n : ℕ)
(m : M₁)
:
@[simp]
theorem
ContinuousLinearMap.ofNat_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
(n : ℕ)
[n.AtLeastTwo]
(m : M₁)
:
def
ContinuousLinearMap.homeomorphOfUnit
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(T : (M₁ →L[R₁] M₁)ˣ)
:
Construct a homeomorphism from an invertible continuous linear map.
Equations
- ContinuousLinearMap.homeomorphOfUnit T = { toFun := ⇑↑T, invFun := ⇑↑T⁻¹, left_inv := ⋯, right_inv := ⋯, continuous_toFun := ⋯, continuous_invFun := ⋯ }
Instances For
@[simp]
theorem
ContinuousLinearMap.homeomorphOfUnit_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(T : (M₁ →L[R₁] M₁)ˣ)
(a : M₁)
:
@[simp]
theorem
ContinuousLinearMap.homeomorphOfUnit_symm_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
(T : (M₁ →L[R₁] M₁)ˣ)
(a : M₁)
:
theorem
ContinuousLinearMap.isHomeomorph_of_isUnit
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{T : M₁ →L[R₁] M₁}
(hT : IsUnit T)
:
IsHomeomorph ⇑T
@[implicit_reducible]
instance
ContinuousLinearMap.applyModule
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
The tautological action by M₁ →L[R₁] M₁ on M.
This generalizes Function.End.applyMulAction.
@[simp]
theorem
ContinuousLinearMap.smul_def
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
(f : M₁ →L[R₁] M₁)
(a : M₁)
:
instance
ContinuousLinearMap.applyFaithfulSMul
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
FaithfulSMul (M₁ →L[R₁] M₁) M₁
ContinuousLinearMap.applyModule is faithful.
instance
ContinuousLinearMap.applySMulCommClass
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
SMulCommClass R₁ (M₁ →L[R₁] M₁) M₁
instance
ContinuousLinearMap.applySMulCommClass'
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
SMulCommClass (M₁ →L[R₁] M₁) R₁ M₁
instance
ContinuousLinearMap.continuousConstSMul_apply
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[ContinuousAdd M₁]
:
ContinuousConstSMul (M₁ →L[R₁] M₁) M₁
theorem
ContinuousLinearMap.isClosed_ker
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₂ M₂]
[T1Space M₂]
(f : M₁ →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.isComplete_ker
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{M' : Type u_9}
[UniformSpace M']
[CompleteSpace M']
[AddCommMonoid M']
[Module R₁ M']
[T1Space M₂]
(f : M' →SL[σ₁₂] M₂)
:
IsComplete ↑(↑f).ker
instance
ContinuousLinearMap.completeSpace_ker
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{M' : Type u_9}
[UniformSpace M']
[CompleteSpace M']
[AddCommMonoid M']
[Module R₁ M']
[T1Space M₂]
(f : M' →SL[σ₁₂] M₂)
:
CompleteSpace ↥(↑f).ker
instance
ContinuousLinearMap.completeSpace_eqLocus
{R₁ : Type u_1}
{R₂ : Type u_2}
[Semiring R₁]
[Semiring R₂]
{σ₁₂ : R₁ →+* R₂}
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{M' : Type u_9}
[UniformSpace M']
[CompleteSpace M']
[AddCommMonoid M']
[Module R₁ M']
[T2Space M₂]
(f g : M' →SL[σ₁₂] M₂)
:
CompleteSpace ↥(LinearMap.eqLocus f g)
def
ContinuousLinearMap.smulRight
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{R : Type u_9}
{S : Type u_10}
[Semiring R]
[Semiring S]
[Module R M₁]
[Module R M₂]
[Module R S]
[Module S M₂]
[IsScalarTower R S M₂]
[TopologicalSpace S]
[ContinuousSMul S M₂]
(c : M₁ →L[R] S)
(f : M₂)
:
The linear map fun x => c x • f. Associates to a scalar-valued linear map and an element of
M₂ the M₂-valued linear map obtained by multiplying the two (a.k.a. tensoring by M₂).
See also ContinuousLinearMap.smulRightₗ and ContinuousLinearMap.smulRightL.
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_smulRight
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{R : Type u_9}
{S : Type u_10}
[Semiring R]
[Semiring S]
[Module R M₁]
[Module R M₂]
[Module R S]
[Module S M₂]
[IsScalarTower R S M₂]
[TopologicalSpace S]
[ContinuousSMul S M₂]
(c : M₁ →L[R] S)
(f : M₂)
:
@[simp]
theorem
ContinuousLinearMap.smulRight_apply
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{R : Type u_9}
{S : Type u_10}
[Semiring R]
[Semiring S]
[Module R M₁]
[Module R M₂]
[Module R S]
[Module S M₂]
[IsScalarTower R S M₂]
[TopologicalSpace S]
[ContinuousSMul S M₂]
{c : M₁ →L[R] S}
{f : M₂}
{x : M₁}
:
@[simp]
theorem
ContinuousLinearMap.smulRight_zero
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{R : Type u_9}
{S : Type u_10}
[Semiring R]
[Semiring S]
[Module R M₁]
[Module R M₂]
[Module R S]
[Module S M₂]
[IsScalarTower R S M₂]
[TopologicalSpace S]
[ContinuousSMul S M₂]
(f : M₁ →L[R] S)
:
@[simp]
theorem
ContinuousLinearMap.zero_smulRight
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{R : Type u_9}
{S : Type u_10}
[Semiring R]
[Semiring S]
[Module R M₁]
[Module R M₂]
[Module R S]
[Module S M₂]
[IsScalarTower R S M₂]
[TopologicalSpace S]
[ContinuousSMul S M₂]
{x : M₂}
:
theorem
ContinuousLinearMap.smulRight_comp_smulRight
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
{M₃ : Type u_9}
[AddCommMonoid M₃]
[Module R₁ M₃]
[TopologicalSpace M₃]
[ContinuousSMul R₁ M₃]
(f : M₃ →L[R₁] R₁)
(g : M₁ →L[R₁] R₁)
{x : M₂}
{y : M₃}
:
@[deprecated ContinuousLinearMap.smulRight_comp_smulRight (since := "2025-12-18")]
theorem
ContinuousLinearMap.smulRight_comp
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
{M₃ : Type u_9}
[AddCommMonoid M₃]
[Module R₁ M₃]
[TopologicalSpace M₃]
[ContinuousSMul R₁ M₃]
(f : M₃ →L[R₁] R₁)
(g : M₁ →L[R₁] R₁)
{x : M₂}
{y : M₃}
:
theorem
ContinuousLinearMap.range_smulRight_apply
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
{R : Type u_9}
[DivisionSemiring R]
[Module R M₁]
[Module R M₂]
[TopologicalSpace R]
[ContinuousSMul R M₂]
{f : M₁ →L[R] R}
(hf : f ≠ 0)
(x : M₂)
:
def
ContinuousLinearMap.toSpanSingleton
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
(x : M₁)
:
Given an element x of a topological space M over a semiring R, the natural continuous
linear map from R to M by taking multiples of x.
Equations
- ContinuousLinearMap.toSpanSingleton R₁ x = { toLinearMap := LinearMap.toSpanSingleton R₁ M₁ x, cont := ⋯ }
Instances For
@[simp]
theorem
ContinuousLinearMap.toSpanSingleton_apply
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
(x : M₁)
(r : R₁)
:
@[simp]
theorem
ContinuousLinearMap.toSpanSingleton_zero
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
:
theorem
ContinuousLinearMap.toSpanSingleton_apply_one
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
(x : M₁)
:
@[deprecated ContinuousLinearMap.toSpanSingleton_apply_one (since := "2025-12-05")]
theorem
ContinuousLinearMap.toSpanSingleton_one
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
(x : M₁)
:
@[simp]
theorem
ContinuousLinearMap.toSpanSingleton_apply_map_one
(R₁ : Type u_1)
[Semiring R₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
(c : R₁ →L[R₁] M₂)
:
@[deprecated ContinuousLinearMap.toSpanSingleton_apply_map_one (since := "2025-12-18")]
theorem
ContinuousLinearMap.smulRight_one_one
(R₁ : Type u_1)
[Semiring R₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
(c : R₁ →L[R₁] M₂)
:
theorem
ContinuousLinearMap.toSpanSingleton_add
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
[ContinuousAdd M₁]
(x y : M₁)
:
theorem
ContinuousLinearMap.toSpanSingleton_smul
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
{α : Type u_9}
[Monoid α]
[DistribMulAction α M₁]
[ContinuousConstSMul α M₁]
[SMulCommClass R₁ α M₁]
(c : α)
(x : M₁)
:
theorem
ContinuousLinearMap.smulRight_id
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
:
theorem
ContinuousLinearMap.smulRight_one_eq_toSpanSingleton
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
(x : M₁)
:
@[deprecated ContinuousLinearMap.smulRight_one_eq_toSpanSingleton (since := "2025-12-05")]
theorem
ContinuousLinearMap.one_smulRight_eq_toSpanSingleton
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
(x : M₁)
:
Alias of ContinuousLinearMap.smulRight_one_eq_toSpanSingleton.
@[simp]
theorem
ContinuousLinearMap.toLinearMap_toSpanSingleton
(R₁ : Type u_1)
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₁]
(x : M₁)
:
theorem
ContinuousLinearMap.comp_toSpanSingleton
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
[ContinuousSMul R₁ M₁]
(f : M₁ →L[R₁] M₂)
(x : M₁)
:
theorem
ContinuousLinearMap.toSpanSingleton_comp
{R₁ : Type u_1}
[Semiring R₁]
{M₁ : Type u_4}
[TopologicalSpace M₁]
[AddCommMonoid M₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₁]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
(f : M₁ →L[R₁] R₁)
(g : M₂)
:
@[simp]
theorem
ContinuousLinearMap.toSpanSingleton_inj
{R₁ : Type u_1}
[Semiring R₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
{f f' : M₂}
:
@[deprecated ContinuousLinearMap.toSpanSingleton_inj (since := "2025-12-18")]
theorem
ContinuousLinearMap.smulRight_one_eq_iff
{R₁ : Type u_1}
[Semiring R₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
{f f' : M₂}
:
Alias of ContinuousLinearMap.toSpanSingleton_inj.
theorem
ContinuousLinearMap.toSpanSingleton_comp_toSpanSingleton
{R₁ : Type u_1}
[Semiring R₁]
{M₂ : Type u_6}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₁ M₂]
[TopologicalSpace R₁]
[ContinuousSMul R₁ M₂]
[ContinuousMul R₁]
{x : M₂}
{c : R₁}
:
theorem
ContinuousLinearMap.map_neg
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
(f : M →SL[σ₁₂] M₂)
(x : M)
:
theorem
ContinuousLinearMap.map_sub
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
(f : M →SL[σ₁₂] M₂)
(x y : M)
:
@[simp]
theorem
ContinuousLinearMap.sub_apply'
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
(f g : M →SL[σ₁₂] M₂)
(x : M)
:
@[implicit_reducible]
instance
ContinuousLinearMap.neg
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
:
@[simp]
theorem
ContinuousLinearMap.neg_apply
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f : M →SL[σ₁₂] M₂)
(x : M)
:
@[simp]
theorem
ContinuousLinearMap.coe_neg
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f : M →SL[σ₁₂] M₂)
:
theorem
ContinuousLinearMap.coe_neg'
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_neg
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f : M →SL[σ₁₂] M₂)
:
@[implicit_reducible]
instance
ContinuousLinearMap.sub
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
:
@[implicit_reducible]
instance
ContinuousLinearMap.addCommGroup
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
:
AddCommGroup (M →SL[σ₁₂] M₂)
Equations
- One or more equations did not get rendered due to their size.
theorem
ContinuousLinearMap.sub_apply
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f g : M →SL[σ₁₂] M₂)
(x : M)
:
@[simp]
theorem
ContinuousLinearMap.coe_sub
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f g : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.coe_sub'
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f g : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.toContinuousAddMonoidHom_sub
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
[Module R M]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[IsTopologicalAddGroup M₂]
(f g : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.comp_neg
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{R₃ : Type u_3}
[Ring R₃]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
{M₃ : Type u_6}
[TopologicalSpace M₃]
[AddCommGroup M₃]
[Module R M]
[Module R₂ M₂]
[Module R₃ M₃]
{σ₁₂ : R →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R →+* R₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[IsTopologicalAddGroup M₂]
[IsTopologicalAddGroup M₃]
(g : M₂ →SL[σ₂₃] M₃)
(f : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.neg_comp
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{R₃ : Type u_3}
[Ring R₃]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
{M₃ : Type u_6}
[TopologicalSpace M₃]
[AddCommGroup M₃]
[Module R M]
[Module R₂ M₂]
[Module R₃ M₃]
{σ₁₂ : R →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R →+* R₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[IsTopologicalAddGroup M₃]
(g : M₂ →SL[σ₂₃] M₃)
(f : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.comp_sub
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{R₃ : Type u_3}
[Ring R₃]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
{M₃ : Type u_6}
[TopologicalSpace M₃]
[AddCommGroup M₃]
[Module R M]
[Module R₂ M₂]
[Module R₃ M₃]
{σ₁₂ : R →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R →+* R₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[IsTopologicalAddGroup M₂]
[IsTopologicalAddGroup M₃]
(g : M₂ →SL[σ₂₃] M₃)
(f₁ f₂ : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.sub_comp
{R : Type u_1}
[Ring R]
{R₂ : Type u_2}
[Ring R₂]
{R₃ : Type u_3}
[Ring R₃]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
{M₂ : Type u_5}
[TopologicalSpace M₂]
[AddCommGroup M₂]
{M₃ : Type u_6}
[TopologicalSpace M₃]
[AddCommGroup M₃]
[Module R M]
[Module R₂ M₂]
[Module R₃ M₃]
{σ₁₂ : R →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R →+* R₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[IsTopologicalAddGroup M₃]
(g₁ g₂ : M₂ →SL[σ₂₃] M₃)
(f : M →SL[σ₁₂] M₂)
:
@[implicit_reducible]
instance
ContinuousLinearMap.ring
{R : Type u_1}
[Ring R]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
[Module R M]
[IsTopologicalAddGroup M]
:
Equations
- One or more equations did not get rendered due to their size.
@[simp]
theorem
ContinuousLinearMap.intCast_apply
{R : Type u_1}
[Ring R]
{M : Type u_4}
[TopologicalSpace M]
[AddCommGroup M]
[Module R M]
[IsTopologicalAddGroup M]
(z : ℤ)
(m : M)
:
theorem
ContinuousLinearMap.toSpanSingleton_pow
{R : Type u_1}
[Ring R]
[TopologicalSpace R]
[IsTopologicalRing R]
(c : R)
(n : ℕ)
:
@[deprecated ContinuousLinearMap.toSpanSingleton_pow (since := "2025-12-18")]
theorem
ContinuousLinearMap.smulRight_one_pow
{R : Type u_1}
[Ring R]
[TopologicalSpace R]
[IsTopologicalRing R]
(c : R)
(n : ℕ)
:
Alias of ContinuousLinearMap.toSpanSingleton_pow.
theorem
ContinuousLinearMap.isOpenMap_of_ne_zero
{R : Type u_1}
{M : Type u_2}
[TopologicalSpace R]
[DivisionRing R]
[ContinuousSub R]
[AddCommGroup M]
[TopologicalSpace M]
[ContinuousAdd M]
[Module R M]
[ContinuousSMul R M]
(f : StrongDual R M)
(hf : f ≠ 0)
:
IsOpenMap ⇑f
A nonzero continuous linear functional is open.
@[simp]
theorem
ContinuousLinearMap.smul_comp
{R : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
{S₃ : Type u_5}
[Semiring R]
[Semiring R₂]
[Semiring R₃]
[Monoid S₃]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{M₂ : Type u_7}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{M₃ : Type u_8}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₃ M₃]
[DistribMulAction S₃ M₃]
[SMulCommClass R₃ S₃ M₃]
[ContinuousConstSMul S₃ M₃]
{σ₁₂ : R →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R →+* R₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
(c : S₃)
(h : M₂ →SL[σ₂₃] M₃)
(f : M →SL[σ₁₂] M₂)
:
@[simp]
theorem
ContinuousLinearMap.comp_smul
{R : Type u_1}
{S : Type u_4}
[Semiring R]
[Monoid S]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{N₂ : Type u_9}
[TopologicalSpace N₂]
[AddCommMonoid N₂]
[Module R N₂]
{N₃ : Type u_10}
[TopologicalSpace N₃]
[AddCommMonoid N₃]
[Module R N₃]
[DistribMulAction S N₃]
[SMulCommClass R S N₃]
[ContinuousConstSMul S N₃]
[DistribMulAction S N₂]
[ContinuousConstSMul S N₂]
[SMulCommClass R S N₂]
[LinearMap.CompatibleSMul N₂ N₃ S R]
(hₗ : N₂ →L[R] N₃)
(c : S)
(fₗ : M →L[R] N₂)
:
@[simp]
theorem
ContinuousLinearMap.comp_smulₛₗ
{R : Type u_1}
{R₂ : Type u_2}
{R₃ : Type u_3}
[Semiring R]
[Semiring R₂]
[Semiring R₃]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{M₂ : Type u_7}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{M₃ : Type u_8}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₃ M₃]
{σ₁₂ : R →+* R₂}
{σ₂₃ : R₂ →+* R₃}
{σ₁₃ : R →+* R₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[SMulCommClass R₂ R₂ M₂]
[SMulCommClass R₃ R₃ M₃]
[ContinuousConstSMul R₂ M₂]
[ContinuousConstSMul R₃ M₃]
(h : M₂ →SL[σ₂₃] M₃)
(c : R₂)
(f : M →SL[σ₁₂] M₂)
:
@[implicit_reducible]
instance
ContinuousLinearMap.distribMulAction
{R : Type u_1}
{R₂ : Type u_2}
{S₃ : Type u_5}
[Semiring R]
[Semiring R₂]
[Monoid S₃]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{M₂ : Type u_7}
[TopologicalSpace M₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{σ₁₂ : R →+* R₂}
[DistribMulAction S₃ M₂]
[ContinuousConstSMul S₃ M₂]
[SMulCommClass R₂ S₃ M₂]
[ContinuousAdd M₂]
:
DistribMulAction S₃ (M →SL[σ₁₂] M₂)
Equations
- ContinuousLinearMap.distribMulAction = { toMulAction := ContinuousLinearMap.mulAction, smul_zero := ⋯, smul_add := ⋯ }
@[implicit_reducible]
instance
ContinuousLinearMap.module
{R : Type u_1}
{R₃ : Type u_3}
{S₃ : Type u_5}
[Semiring R]
[Semiring R₃]
[Semiring S₃]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{M₃ : Type u_8}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₃ M₃]
[Module S₃ M₃]
[SMulCommClass R₃ S₃ M₃]
[ContinuousConstSMul S₃ M₃]
{σ₁₃ : R →+* R₃}
[ContinuousAdd M₃]
:
Equations
- ContinuousLinearMap.module = { toDistribMulAction := ContinuousLinearMap.distribMulAction, add_smul := ⋯, zero_smul := ⋯ }
instance
ContinuousLinearMap.isCentralScalar
{R : Type u_1}
{R₃ : Type u_3}
{S₃ : Type u_5}
[Semiring R]
[Semiring R₃]
[Semiring S₃]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{M₃ : Type u_8}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₃ M₃]
[Module S₃ M₃]
[SMulCommClass R₃ S₃ M₃]
[ContinuousConstSMul S₃ M₃]
{σ₁₃ : R →+* R₃}
[Module S₃ᵐᵒᵖ M₃]
[IsCentralScalar S₃ M₃]
:
IsCentralScalar S₃ (M →SL[σ₁₃] M₃)
def
ContinuousLinearMap.coeLM
{R : Type u_1}
(S : Type u_4)
[Semiring R]
[Semiring S]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{N₃ : Type u_10}
[TopologicalSpace N₃]
[AddCommMonoid N₃]
[Module R N₃]
[Module S N₃]
[SMulCommClass R S N₃]
[ContinuousConstSMul S N₃]
[ContinuousAdd N₃]
:
The coercion from M →L[R] M₂ to M →ₗ[R] M₂, as a linear map.
Equations
- ContinuousLinearMap.coeLM S = { toFun := ContinuousLinearMap.toLinearMap, map_add' := ⋯, map_smul' := ⋯ }
Instances For
@[simp]
theorem
ContinuousLinearMap.coeLM_apply
{R : Type u_1}
(S : Type u_4)
[Semiring R]
[Semiring S]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{N₃ : Type u_10}
[TopologicalSpace N₃]
[AddCommMonoid N₃]
[Module R N₃]
[Module S N₃]
[SMulCommClass R S N₃]
[ContinuousConstSMul S N₃]
[ContinuousAdd N₃]
(self : M →L[R] N₃)
:
def
ContinuousLinearMap.coeLMₛₗ
{R : Type u_1}
{R₃ : Type u_3}
{S₃ : Type u_5}
[Semiring R]
[Semiring R₃]
[Semiring S₃]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{M₃ : Type u_8}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₃ M₃]
[Module S₃ M₃]
[SMulCommClass R₃ S₃ M₃]
[ContinuousConstSMul S₃ M₃]
(σ₁₃ : R →+* R₃)
[ContinuousAdd M₃]
:
The coercion from M →SL[σ] M₂ to M →ₛₗ[σ] M₂, as a linear map.
Equations
- ContinuousLinearMap.coeLMₛₗ σ₁₃ = { toFun := ContinuousLinearMap.toLinearMap, map_add' := ⋯, map_smul' := ⋯ }
Instances For
@[simp]
theorem
ContinuousLinearMap.coeLMₛₗ_apply
{R : Type u_1}
{R₃ : Type u_3}
{S₃ : Type u_5}
[Semiring R]
[Semiring R₃]
[Semiring S₃]
{M : Type u_6}
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
{M₃ : Type u_8}
[TopologicalSpace M₃]
[AddCommMonoid M₃]
[Module R₃ M₃]
[Module S₃ M₃]
[SMulCommClass R₃ S₃ M₃]
[ContinuousConstSMul S₃ M₃]
(σ₁₃ : R →+* R₃)
[ContinuousAdd M₃]
(self : M →SL[σ₁₃] M₃)
:
def
ContinuousLinearMap.lcomp
{R : Type u_1}
{U : Type u_2}
{V : Type u_3}
(W : Type u_4)
[CommSemiring R]
[AddCommMonoid U]
[Module R U]
[TopologicalSpace U]
[AddCommMonoid V]
[Module R V]
[TopologicalSpace V]
[AddCommMonoid W]
[Module R W]
[TopologicalSpace W]
[ContinuousAdd W]
[ContinuousConstSMul R W]
(f : U →L[R] V)
:
Composition of continuous linear maps, as a linear map. Compare LinearMap.lcomp.
Equations
Instances For
@[simp]
theorem
ContinuousLinearMap.lcomp_apply
{R : Type u_1}
{U : Type u_2}
{V : Type u_3}
(W : Type u_4)
[CommSemiring R]
[AddCommMonoid U]
[Module R U]
[TopologicalSpace U]
[AddCommMonoid V]
[Module R V]
[TopologicalSpace V]
[AddCommMonoid W]
[Module R W]
[TopologicalSpace W]
[ContinuousAdd W]
[ContinuousConstSMul R W]
(f : U →L[R] V)
(l : V →L[R] W)
:
def
ContinuousLinearMap.llcomp
(R : Type u_1)
(U : Type u_2)
(V : Type u_3)
(W : Type u_4)
[CommSemiring R]
[AddCommMonoid U]
[Module R U]
[TopologicalSpace U]
[AddCommMonoid V]
[Module R V]
[TopologicalSpace V]
[ContinuousAdd V]
[ContinuousConstSMul R V]
[AddCommMonoid W]
[Module R W]
[TopologicalSpace W]
[ContinuousAdd W]
[ContinuousConstSMul R W]
:
Composition of continuous linear maps, as a bilinear map. Compare LinearMap.llcomp.
Equations
- ContinuousLinearMap.llcomp R U V W = { toFun := fun (l : U →L[R] V) => ContinuousLinearMap.lcomp W l, map_add' := ⋯, map_smul' := ⋯ }
Instances For
@[simp]
theorem
ContinuousLinearMap.llcomp_apply
(R : Type u_1)
(U : Type u_2)
(V : Type u_3)
(W : Type u_4)
[CommSemiring R]
[AddCommMonoid U]
[Module R U]
[TopologicalSpace U]
[AddCommMonoid V]
[Module R V]
[TopologicalSpace V]
[ContinuousAdd V]
[ContinuousConstSMul R V]
[AddCommMonoid W]
[Module R W]
[TopologicalSpace W]
[ContinuousAdd W]
[ContinuousConstSMul R W]
(l : U →L[R] V)
:
def
ContinuousLinearMap.toSpanSingletonLE
(R : Type u_1)
(S : Type u_2)
(M : Type u_3)
[Semiring R]
[Semiring S]
[AddCommMonoid M]
[Module R M]
[Module S M]
[SMulCommClass R S M]
[TopologicalSpace M]
[ContinuousAdd M]
[ContinuousConstSMul S M]
[TopologicalSpace R]
[ContinuousSMul R M]
:
ContinuousLinearMap.toSpanSingleton as a linear equivalence.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
ContinuousLinearMap.toSpanSingletonLE_apply
(R : Type u_1)
(S : Type u_2)
(M : Type u_3)
[Semiring R]
[Semiring S]
[AddCommMonoid M]
[Module R M]
[Module S M]
[SMulCommClass R S M]
[TopologicalSpace M]
[ContinuousAdd M]
[ContinuousConstSMul S M]
[TopologicalSpace R]
[ContinuousSMul R M]
:
@[simp]
theorem
ContinuousLinearMap.toSpanSingletonLE_symm_apply
(R : Type u_1)
(S : Type u_2)
(M : Type u_3)
[Semiring R]
[Semiring S]
[AddCommMonoid M]
[Module R M]
[Module S M]
[SMulCommClass R S M]
[TopologicalSpace M]
[ContinuousAdd M]
[ContinuousConstSMul S M]
[TopologicalSpace R]
[ContinuousSMul R M]
:
def
ContinuousLinearMap.smulRightₗ
{R : Type u_1}
{S : Type u_2}
{T : Type u_3}
{M : Type u_4}
{M₂ : Type u_5}
[Semiring R]
[Semiring S]
[Semiring T]
[Module R S]
[AddCommMonoid M₂]
[Module R M₂]
[Module S M₂]
[IsScalarTower R S M₂]
[TopologicalSpace S]
[TopologicalSpace M₂]
[ContinuousSMul S M₂]
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
[ContinuousAdd M₂]
[Module T M₂]
[ContinuousConstSMul T M₂]
[SMulCommClass R T M₂]
[SMulCommClass S T M₂]
(c : M →L[R] S)
:
Given c : E →L[R] S, c.smulRightₗ is the linear map from F to E →L[R] F
sending f to fun e => c e • f. See also ContinuousLinearMap.smulRightL.
Equations
- c.smulRightₗ = { toFun := c.smulRight, map_add' := ⋯, map_smul' := ⋯ }
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_smulRightₗ
{R : Type u_1}
{S : Type u_2}
{T : Type u_3}
{M : Type u_4}
{M₂ : Type u_5}
[Semiring R]
[Semiring S]
[Semiring T]
[Module R S]
[AddCommMonoid M₂]
[Module R M₂]
[Module S M₂]
[IsScalarTower R S M₂]
[TopologicalSpace S]
[TopologicalSpace M₂]
[ContinuousSMul S M₂]
[TopologicalSpace M]
[AddCommMonoid M]
[Module R M]
[ContinuousAdd M₂]
[Module T M₂]
[ContinuousConstSMul T M₂]
[SMulCommClass R T M₂]
[SMulCommClass S T M₂]
(c : M →L[R] S)
:
@[implicit_reducible]
instance
ContinuousLinearMap.algebra
{R : Type u_1}
{S : Type u_2}
{M : Type u_3}
[Semiring R]
[TopologicalSpace M]
[AddCommGroup M]
[Module R M]
[CommSemiring S]
[Module S M]
[SMulCommClass R S M]
[SMul S R]
[IsScalarTower S R M]
[ContinuousConstSMul S M]
[IsTopologicalAddGroup M]
:
Equations
@[simp]
theorem
ContinuousLinearMap.algebraMap_apply
{R : Type u_1}
{S : Type u_2}
{M : Type u_3}
[Semiring R]
[TopologicalSpace M]
[AddCommGroup M]
[Module R M]
[CommSemiring S]
[Module S M]
[SMulCommClass R S M]
[SMul S R]
[IsScalarTower S R M]
[ContinuousConstSMul S M]
[IsTopologicalAddGroup M]
(r : S)
(m : M)
:
def
topDualPairing
(𝕜 : Type u_1)
(E : Type u_2)
[CommSemiring 𝕜]
[TopologicalSpace 𝕜]
[ContinuousAdd 𝕜]
[AddCommMonoid E]
[Module 𝕜 E]
[TopologicalSpace E]
[ContinuousConstSMul 𝕜 𝕜]
:
The canonical pairing of a vector space and its topological dual.
Equations
Instances For
@[simp]
theorem
topDualPairing_apply
{𝕜 : Type u_1}
{E : Type u_2}
[CommSemiring 𝕜]
[TopologicalSpace 𝕜]
[ContinuousAdd 𝕜]
[AddCommMonoid E]
[Module 𝕜 E]
[TopologicalSpace E]
[ContinuousConstSMul 𝕜 𝕜]
(v : E →L[𝕜] 𝕜)
(x : E)
: